L(s) = 1 | − 0.863·2-s − 1.78·3-s − 1.25·4-s + 1.53·6-s − 4.46·7-s + 2.81·8-s + 0.179·9-s − 4.87·11-s + 2.23·12-s − 4.21·13-s + 3.85·14-s + 0.0812·16-s + 4.42·17-s − 0.155·18-s − 7.05·19-s + 7.96·21-s + 4.21·22-s − 6.18·23-s − 5.01·24-s + 3.63·26-s + 5.02·27-s + 5.60·28-s + 7.05·29-s + 4.29·31-s − 5.69·32-s + 8.69·33-s − 3.82·34-s + ⋯ |
L(s) = 1 | − 0.610·2-s − 1.02·3-s − 0.627·4-s + 0.628·6-s − 1.68·7-s + 0.993·8-s + 0.0598·9-s − 1.47·11-s + 0.645·12-s − 1.16·13-s + 1.03·14-s + 0.0203·16-s + 1.07·17-s − 0.0365·18-s − 1.61·19-s + 1.73·21-s + 0.898·22-s − 1.28·23-s − 1.02·24-s + 0.713·26-s + 0.967·27-s + 1.05·28-s + 1.30·29-s + 0.771·31-s − 1.00·32-s + 1.51·33-s − 0.655·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 0.863T + 2T^{2} \) |
| 3 | \( 1 + 1.78T + 3T^{2} \) |
| 7 | \( 1 + 4.46T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 + 6.18T + 23T^{2} \) |
| 29 | \( 1 - 7.05T + 29T^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 - 2.88T + 37T^{2} \) |
| 41 | \( 1 - 8.91T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 9.41T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 - 0.904T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 6.12T + 89T^{2} \) |
| 97 | \( 1 + 2.18T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.023223414364032997634015842396, −6.91545649074271437378951058332, −6.29045799618453617812996468126, −5.65088294595298000920230407889, −4.91991014245730336669869830064, −4.24673267128685509732673411095, −3.10736795338143070047570432645, −2.35579887695312022712027636684, −0.62251088924846565022089346055, 0,
0.62251088924846565022089346055, 2.35579887695312022712027636684, 3.10736795338143070047570432645, 4.24673267128685509732673411095, 4.91991014245730336669869830064, 5.65088294595298000920230407889, 6.29045799618453617812996468126, 6.91545649074271437378951058332, 8.023223414364032997634015842396